
as claimed provided \(d\ge 3\) . The main gist of [1]} is now to remove the \(L^2\) -piece from this argument. This is subtle, as the free estimate involved \((t-s)^{-\frac{d}{2}}\)  which is not integrable at \(s=t\) . To overcome this difficulty, Journé,
Soffer, and Sogge used the bound
\(\sup _{1\le p\le \infty }\Vert  e^{-it\Delta } V e^{it\Delta } \Vert _{p\rightarrow p} \le \Vert \hat{V}\Vert _1\) 
